potential function in a continuous dissipative chaotic ... · potential function in a continuous...
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Potential Function in a Continuous Dissipative Chaotic System:
Decomposition Scheme and Role of Strange AttractorYian Ma,1 Qijun Tan,2 Ruoshi Yuan,1 Bo Yuan,1, a) and Ping Ao3, b)1)Department of Computer Science and Engineering, Shanghai Jiao Tong University2)School of Mathematical Sciences, Fudan University, Shanghai, 200433, China3)Shanghai Center for Systems Biomedicine and Department of Physics
Shanghai Jiao Tong University, Shanghai, 200240, China
(Dated: 4 June 2018)
In this paper, we demonstrate, first in literature known to us, that potential functions can be constructed incontinuous dissipative chaotic systems and can be used to reveal their dynamical properties. To attain thisaim, a Lorenz-like system is proposed and rigorously proved chaotic for exemplified analysis. We explicitlyconstruct a potential function monotonically decreasing along the system’s dynamics, revealing the structureof the chaotic strange attractor. The potential function can have different forms of construction. We alsodecompose the dynamical system to explain for the different origins of chaotic attractor and strange attrac-tor. Consequently, reasons for the existence of both chaotic nonstrange attractors and nonchaotic strangeattractors are clearly discussed within current decomposition framework.
Potential function (also known as energy func-tion, generalized Hamiltonian, or Lyapunov func-tion, under different contexts) describes nonlin-ear dynamical system from a global point of view.Along this scalar function, all the states in phasespace move downward. Consequently, the poten-tial function accounts for both detailed structureand long term trend of the system’s dynamics, in-dicating its performance and stability at the sametime. It is also anticipated that when the poten-tial function of a system becomes constant, thesystem has evolved into an “attractor”. Hence,potential function has special theoretical impor-tance to chaotic system in that it helps reveal thecomplex structure of chaotic attractor. Unfortu-nately, fundamental difficulties pertaining to itsconstruction in nonlinear dynamical systems stillneed solution. Failure in its construction has evenprompted claims that potential function does notexist in complex systems1,2. In this paper, wedemonstrate that potential functions can be con-structed in chaotic dynamical systems. We firstpresent a simplified geometrical Lorenz attrac-tor, and rigorously prove that it is chaotic by itsPoincare map (which is itself an effort interest-ing to many researchers3). Then we analyticallyconstruct a potential function for the system, ac-counting for the structure of the chaotic attrac-tor. With this potential function, we discoverthat chaotic attractor may not be a strange at-tractor and vice versa. This corresponds to pre-vious observations and is explained in detail byvirtue of our constructive approach.
a)Electronic mail: [email protected])Electronic mail: [email protected].
I. INTRODUCTION
Nonlinear dynamics underlying many natural andtechnological systems is described by a set of ordinarydifferential equations:
dx(t)
dt= f(x). (1)
This description defines a fixed rule dictating the trendof evolution that a current state will follow into its im-mediate future. Overtime, systems with even simple de-terministic rules can give rise to seemingly “fortuitous”phenomena4, generally known as chaos. It remains anintriguing problem as to how these phenomena can beanalyzed in a global sense5, under the light of a genericframework in nonequilibrium dynamics6. Ideally, such ageneric framework should account for: a unified (proba-bly geometric) structure underlying the equations of evo-lution, a comparable measure of the system’s differentstates, and an accurate reflection of the dynamic processgenerated by the system.On an historical account, active search for such a
generic framework begins in the 1970s, when Rene Thomand Christopher Zeeman proposed that potential func-tions and their deformation7 can describe “the evolutionof form in all aspects of nature, and hence it embodiesa theory of great generality”8. Thom’s proposal is inge-nious. Because such a scalar function incorporates thepreviously mentioned attributes into one single quantity:potential function. It not only generalizes existing ap-proaches of Lyapunov function and first integral, but alsoencompasses concepts like stability9 and reversibility10
into a uniform framework. Even, natural and technologi-cal systems can directly be modeled with potential func-tions so that the behaviors can still be described “evenwhen all the internal parameters describing the systemare not explicitly known”7.However, according to Stephen Smale, Thom’s math-
ematical approach “deals in with only a few known
2
examples”11, and hence lacks practical effectiveness.What Thom and his followers failed to obtain is a proofof existence or even a method of construction for poten-tial functions in complex systems, such as systems withoscillation or chaotic behaviors. Frustration has evenprompted claims that potential functions do not existin the situations of complex dynamical systems1,2
Previously, we have rigorously defined potential func-tion in mathematical terms (see definition 1) and al-ready demonstrated that potential functions (or Lya-punov functions) can be analytically constructed in os-cillating systems12,13. In this paper, we further motivatesuch research by showing the construction of potentialfunctions in chaotic systems, and providing additionalinsights for chaotic and strange attractors.Actually, constructing potential-like functions in
chaotic systems, functions with a restricted part of theproperties held by potential functions, is an approach al-ready taken by researchers. Until very recently, thereare still various efforts addressing the issue. There aregeneralized Hamiltonian approach14, energy-like func-tion technique15, minimum action method16, and etc.,in search for a unified description of chaotic dynam-ics. These previous methods all construct a potential-like function to analyze some chaotic system such as theLorenz system17. Unfortunately, the scalar functions inthese works all lack certain important properties (see sec-tion 6).Those important drawbacks of the existing methods
motivate a real potential function to describe the behav-ior of chaotic systems. Ideally, it should be a continu-ous function in the phase space, monotonically decreasingwith time. Also, when time approaches infinity, poten-tial function should stabilize to some finite quantity ifthe original system is not divergent. In this paper, wecombine these ideas together as a potential function andtry to obtain it in chaotic systems.The paper is organized as follows. First of all, we
define an ideal potential function for the description ofchaotic systems. To analyze chaotic systems in detailwith this potential function, we create an attractor thatis chaotic by definition. Then, potential function for thischaotic attractor is constructed, showing the structure ofthe chaotic strange attractor. In addition, our frameworkprovides a decomposition of the original vector field. Thedecomposition helps understand the different origins forchaotic attractor and strange attractor, explaining whythere exists both chaotic nonstrange attractors and non-chaotic strange attractors.
II. POTENTIAL FUNCTION
We first state the definition of a potential function,which is a natural description of dynamical systems withmonotonic properties. Then we will discuss a decomposi-tion scheme of generic dynamical systems associated withthe potential function.
Definition 1 (Potential Function18,19). Let Ψ : Rn −→ R
be a continuous function. Then Ψ satisfying the followingcondition is called a potential function for the dynamicalsystem x = f(x) : Rn −→ Rn.
(a) Ψ(x) = dΨ/dt|x 6 0 for all x ∈ Rn if Ψ(x) exists.
(b) ∇Ψ(x∗) = 0 if and only if x∗ ∈ A, where A is theattractor of the dynamical system: x = f(x).
Here, an attractor A of a dynamical system with flowφt can be fixed point, limit cycle, or chaotic attractor. Inthis sense, a potential function is a Lyapunov function.An attractor A is formally defined as the following.
Definition 2 (Attractor). An attractor A of a dynamicalsystem x = f(x) with flow φt is a compact invariant set,with an open set U containing A such that for each x ∈ U ,φt(x) ∈ U for all t > 0 and A =
⋂
t>0 φt(U).
This definition balances between different fashions ofliteratures20,21 and is exactly the same as the definitionof an “attracting set” in a classical textbook of dynamicalsystems22.
A. Decomposition Scheme
Many efforts generalizing Hamiltonian or gradient dy-namics realize that there generally exists a decompositionof dynamical systems x = f(x)19,23,24:
x = f(x) = M(x)∇Ψ(x).
And M(x) can be decomposed into:
M(x) = J(x)−D(x) +Q(x),
where J(x) and D(x) are semi-positive definite symmet-ric matrices and Q(x) is skew-symmetric.In our definition of the potential function Ψ(x), how-
ever, matrix J(x) do not appear. That is:
x = f(x) (2)
= −D(x)∇Ψ(x) +Q(x)∇Ψ(x).
This means that a generic system is only composed ofan energy dissipating (gradient) part and an energy con-served (rotation) part. For the gradient part, potentialΨ is a common energy function; for the rotation part, Ψis a first integral. And we can express these two partsonce we find the potential function for the system19:
D = −f · ∇Ψ
∇Ψ · ∇ΨI, (3)
and
Q =f ×∇Ψ
∇Ψ · ∇Ψ. (4)
3
Here, I denotes identity matrix and the generalized crossproduct of two vectors defines a matrix: x × y = A =(aij)n×n = (xiyj − xjyi)n×n.Clearly, (D∇Ψ(x)) × ∇Ψ = 0 and (Q∇Ψ(x)) ·
∇Ψ = 0, corresponding exactly to the curl-free com-ponent and divergence-free component in Helmholtzdecomposition25.Further, as have been discussed in the context of
nonequilibrium thermal dynamics26, such two parts cor-respond to two different structures in geometry: a dissi-pative bracket {·, ·}; and a generalized Poisson bracket27
[·, ·]28.A dissipative bracket {·, ·} associates with matrix
D(x):
{f, g} = ∂ifDij∂jg,
and is generally defined as symmetric: {f, g} = {g, f};and semi-positive definite: {f, f} > 0; satisfying Leibniz’rule: {fg, h} = f{g, h}+ g{f, h}.While a generalized Poisson bracket [·, ·] associates
with matrix Q(x):
[f, g] = ∂ifQij∂jg,
and can be generally defined as antisymmetric: [f, g] =− [g, f ]; satisfying Leibniz’ rule: [fg, h] = f [g, h]+g[f, h].Hence, the original differential equations can be ex-
pressed as:
xi = −{xi,Ψ}+ [xi,Ψ] .
That is, a generic dynamical system is a direct composi-tion of the two well-studied geometric structures.Later, this gradient-rotation decomposition would pro-
vide additional insight to the understanding of chaoticattractors and strange attractors.
III. SIMPLIFIED GEOMETRIC LORENZ ATTRACTOR
As can be seen in previous works, many efforts havebeen made to analyze Lorenz system17 as a typical modelfor chaos. Yet, to the best knowledge of the authors,there is only numerical evidence29 that the Lorenz equa-tions support a robust strange attractor30. Total un-derstanding of the Lorenz attractor, including but notlimited to an analytic proof that the Lorenz attractor ischaotic is still lacking31.An early work32 attempted to study chaotic systems by
constructing a geometric model in a piecewise fashion toresemble the Lorenz system. The resultant “geometricLorenz attractor” from the piecewise model is studiedin some depth and an analogy is made between it andthe Lorenz system30. This methodology is practicallyeffective, yet the model system can become even simplerto be analytically proved as chaotic.Hence, we start out constructing a simplified geomet-
ric Lorenz attractor. The model system is described
Figure 1 | The Simplified Geometric LorenzAttractor. The dynamical system we study here is definedpiecewisely in region RA, RB , and RC , along with regionRB′ and RC′ as the symmetric counterparts of RB and RC .The front, side, and top view of the regions of definition areshown in panel (a) through panel (c) respectively.Trajectories of this dynamical system would converge intoan attractor AL (see subsection D and figure 7), which is asimplified version of the geometric Lorenz attractor32 .
by piecewise continuous ordinary differential equations(ODE), similar to the “geometric Lorenz attractor”. Weintegrate trajectories in each continuous region of themodel system. Then we reveal the structure of the at-tractor by finding the Poincare map between the continu-ous regions. Through the Poincare map, the attractor isproved to be a chaotic attractor according to the widelyapplied definition22 of Devaney chaos.
A. Model System Description
The piecewise continuous ODE model is described ineach continuous region (from RA to RC , along with RB′
and RC′ as the symmetric counterparts of RB and RC)as follows, corresponding to Figure (1).
1. In region RA, where x ∈ [0, 2]33, y ∈ [−2, 2], z ∈ [0, 2],yz ∈ [−2, 2]:
x = 0y = yz = −z.
(5)
Dynamics in this region is characterized by saddlepoints at y = z = 0. These saddle points are respon-sible for causing bifurcation in originally close trajec-tories.
4
2. Region RB is defined as: x ∈ [0 , 2/3 + 8/(3π) × θ],
y ∈ (2, 4], z ∈ [0, 2],√
(z − 2)2 + (y − 2)2 ∈ [1, 2],where:
θ = arccosy − 2
√
(z − 2)2 + (y − 2)2,
denoting the angle that point (y, z) form with respectto the center (2, 2).
In RB:
x = −x
θ + π/4y = 2− zz = y − 2.
(6)
Trajectories in this region rotate for an angle of π/2with respect to y = z = 2 and contract in the x direc-tion.
3. In region RC , where x ∈ [0, 2/3], y ∈ [−1, 4], z > 2,√
(z − 2)2 + (y − 2)2 > 1,√
(z − 2)2 + (y − 3/2)2 6
5/2:
x = 0y = 2− z
z =9y
8−
21
8+
√
(3y − 7)2 + 8(z − 2)2
8.
(7)
In this region, trajectories rotate for another angle ofπ with respect to y = z = 2 and expand in the ydirection.
The whole system is set symmetrical with respect tothe line: x = 1; y = 0; z ∈ R. So we change thecoordinate of (x, y, z) into (2 − x,−y, z) to have ex-pressions of the vector field in region RB′ and RC′
from expressions in region RB and RC .
4. Region RB′ is defined as: x ∈ [4/3 − 8/(3π) × θ , 2],
y ∈ [−4,−2), z ∈ [0, 2],√
(z − 2)2 + (y + 2)2 ∈ [1, 2],where:
θ = arccos−y − 2
√
(z − 2)2 + (y + 2)2,
denoting the angle that point (y, z) form with respectto the center (−2, 2).
In RB′ :
x =2− x
θ + π/4y = z − 2z = −y − 2.
(8)
Vector field in region RB′ corresponds exactly to thatin RB.
5. In region RC′ , where x ∈ [4/3, 2], y ∈ [−4, 1], z > 2,√
(z − 2)2 + (y + 2)2 > 1,√
(z − 2)2 + (y + 3/2)2 6
5/2:
x = 0y = z − 2
z = −9y
8−
21
8+
√
(3y + 7)2 + 8(z − 2)2
8.
(9)
Vector field in region RC′ corresponds exactly to thatin region RC .
We note that the model system in region RB′ and RC′
is just a change of variables of the system in region RB
and RC . So, to avoid redundancy, we will only takeregion RA, RB and RC to represent all the regions ofdefinition in the following analysis.
B. Near Saddle-Focus Fixed Points
We have constructed the model system containing onesaddle fixed point. As in the Lorenz system, there wouldactually be another two saddle-focus fixed points whenthe system expands to the whole R3 space. Here, wecomplete the dynamical system near the two saddle-focusfixed points so that the convergence behavior away fromthe attractor can be further demonstrated.We denote the regions of definition discussed here as
region RD and RD′ (see Figure 2), each consisting ofthree parts: region RDA , RDB , and RDC (for region RD
as example). Region RD and RD′ are symmetrical withrespect to the line: x = 1; y = 0; z ∈ R, just as inthe previous section. Hence, we follow the conventionstated in the previous section: to take region RDA , RDB ,and RDC representing their symmetrical counterparts.The regions: RDA , RDB , and RDC and the differentialequations in them are written as the following.
1. Region RDA is close to region RA and is defined as:x ∈ [0, 2], y ∈ [1, 2], z ∈ [1, 2], yz ∈ (2, 4]. We simplytake differential dynamical system in it the same asthat in region RA:
x = 0y = yz = −z.
(10)
Hence, states in this region are unstable in the y direc-tion and stable in the z direction, causing a rotationeffect.
2. Region RDB is close to region RB and is defined as:x ∈ [0 , 2/3 + 8/(3π) × θ], y ∈ (2, 3], z ∈ [1, 2],√
(z − 2)2 + (y − 2)2 < 1.
Here,
θ = arccosy − 2
√
(z − 2)2 + (y − 2)2,
5
Figure 2 | Near Saddle-Focus Fixed Points. When thesystem expand to contain region RD and RD′ , twosaddle-focus fixed points would emerge. This figureelaborates on the system near the two saddle-focus fixedpoints. (a), The regions containing the two saddle-focusfixed points, i.e. RD and RD′ are shown along with otherregions. (b-d), The front, top, and side view of region RD
are shown respectively.
denoting the angle that point (y, z) form with respectto the center (2, 2).
In region RDB :
x = −x
θ + π/4y = 2− zz = y − 2.
(11)
Same as in region RB, trajectories in this region rotatefor an angle of π/2 with respect to y = z = 2 andcontract in the x direction.
3. In region RDC , where x ∈ [0, 2/3], y ∈ [1, 3], z > 2,√
(z − 2)2 + (y − 2)2 < 1:
x = 0
y = 2− z + (y − 2)
(
1√
(z − 2)2 + (y − 2)2− 1
)
z = y − 2 + (z − 2)
(
1√
(z − 2)2 + (y − 2)2− 1
)
.
(12)
In this region, trajectories tend to converge to the unit-radius circle centered at y = z = 2. Hence, statesin the whole region RD are attracted to the circle:x = 0,
√
(z − 2)2 + (y − 2)2 = 1.
Here, it is observable that region RD, containing asaddle-focus fixed point, form a semi-stable limit cy-cle at x = 0,
√
(z − 2)2 + (y − 2)2 = 1 (when y, z ∈
Figure 3 | Simulated Trajectory of the System.We simulated a trajectory of the dynamical systemconstructed. It appears to have “erratic” behaviors, similarto the Lorenz system.
[1, 2], the curve of the limit cycle changes expression to:x = 0, yz = 2). This limit cycle locates at the bound-aries between region RD and its adjacent regions. Thisphenomenon corresponds with many observations thatfixed points transit into chaotic behaviors through limitcycles16.
Also, this section demonstrates that the domain of def-inition in the model system is not restricted to the re-gions discussed above. If we take dynamical system inthe rest of R3 space converging into the defined regions(RA through RD, RB′ through RD′), domain of defini-tion can be expanded to the whole space.
C. Trajectory and Poincare Map
Based on Equation (5-9), we simulate the trajectory ofthe dynamical system (shown in Figure (3)). Actually,trajectories in each region can be analytically solved. Tostudy the structure of the attractor, we solve the trajec-tories in region RA, RB , and RC respectively:
1. In region RA, trajectories are represented as:
x = x0
y = y0et
z = z0e−t,
(13)
where z0 can usually be taken as 2.
Hence, states in this region are exponentially unstablein the y direction and exponentially stable in the zdirection.
6
Figure 4 | Poincare Map of the Dynamical System.This Poincare map is taken over the surface of z = 2. It canreadily be observed that this map is a “baker’s map”34 andcreates a Cantor set multiplying a real line segment.
2. In region RB, trajectories are:
x = x0
(
1
3−
4
3πt
)
y =√
y20 + z20 cos t+ 2
z =√
y20 + z20 sin t+ 2,
(14)
where y0 can be 2.
Here, we can observe that x(t) decreases monotoni-cally while y(t) and z(t) form a circle.
3. In region RC , trajectories are:
x = x0
y =√
y20 + z20 cos t+1
3
(
1−√
y20 + z20
)
+ 2
z =√
y20 + z20 sin t+ 2,
(15)
where z0 can be 2.
Hence, trajectories in region RC move along circlesdetermined by initial conditions.
Then, to further study the structure of the attractorof the system, we want to calculate the Poincare map ofthe system. Here, we take Poincare surface of section as:z = 2, and find the resultant Poincare map as a discretedynamical system defined on [0, 2] × [−1, 1] (shown inFigure (4) and follows).When (x, y) ∈ [0, 2]× [0, 1],
{
xn+1 =1
3xn
yn+1 = 2yn − 1;(16)
When (x, y) ∈ [0, 2]× [−1, 0),
{
xn+1 =1
3xn +
4
3yn+1 = 2yn + 1.
(17)
The above discrete dynamical system is just a “baker’smap” defined in literatures before34, and can be under-stood figuratively as the following. In the x direction,the mapping is contractive. The square of definition:[0, 2]× [−1, 1] is contracted to the one third of it, form-ing a rectangle: [0, 2/3]× [−1, 1]. In the y direction, themapping is expansive just as the doubling map22. Therectangle is stretched to: [0, 2/3]× [−3, 1]. Then we keepthe right half of the resulted rectangle and move the lefthalf: [0, 2/3]× [−3,−1) to the position: [0, 2/3]× [−1, 1).It can readily be seen that the invariant set is formedby iteratively removing the middle third of the intervalsalong the x direction.
D. Attractor of the Model System
We denote the attractor of the model system as: AL.And with the Poincare map of the model system definedas a dynamical system on [0, 2]× [−1, 1] (in the previoussection), we denote its attractor as: AP . In this sec-tion, we first express attractor AP of the Poincare mapin terms of the Cantor set; then we express attractor AL
of the original model system.We have already found the Poincare map of the model
system as iteratively removing the middle third of the in-variant sets along the x direction. That is: first, removethe set (2/3, 4/3)× [−1, 1]; then, remove the middle thirdof the left two sets [0, 2/3]× [−1, 1] and [4/3, 2]× [−1, 1];and iterate the process all along. We present all the re-moved intervals iteratively as the following:
C1 =
(
2
3,4
3
)
× [−1, 1] ,
and
Cn+1 =
(
Cn
3
⋃ Cn + 4
3
)
× [−1, 1] . (18)
Then the attractor of the Poincare map is [0, 2]×[−1, 1]minus the union of all the sets Ci:
AP = [0, 2]× [−1, 1]−∞⋃
i=1
Ci × [−1, 1]
=
(
[0, 2]−∞⋃
i=1
Ci
)
× [−1, 1]
= C× [−1, 1] , (19)
where C denotes the Cantor set35 defined on the intervalof [0, 2].Hence, attractor AP of the Poincare map is the Cantor
set multiplying a real line segment. We can calculate itsbox-counting dimension35 to be:
db(AP ) = limǫ→0
logN(ǫ,AP )
log(1/ǫ)= 1 + ln(2)/ln(3). (20)
7
Hence, the attractor AP is of fractal dimension, a strangeattractor36.With the trajectories of the system analytically
solved in each region, we further express attrac-tor AL of the model system as (assuming θ =
arccos (y − 2)/√
(z − 2)2 + (y − 2)2):In region RA, x ∈ C;in region RB, (4/(3π)× θ + 1/3)−1 x ∈ C;in region RC , x ∈ C.The box-counting dimension of attractor AL is then
calculated to be:
db(AL) = limǫ→0
logN(ǫ,AL)
log(1/ǫ)= 2 + ln(2)/ln(3). (21)
It is hence a strange attractor with fractal dimension.It will be proved in the following section that attractors
AP and AL are also chaotic attractors.
E. Proof of the Attractor as Chaotic
By the widely applied definition22 of Devaney chaos,an attractor A is defined as a chaotic attractor if:
1. the attractor is indecomposable (i.e., if ∅ 6= A′ ⊆ A is
an attractor, then A′ = A);
2. the system is sensitive to initial conditions when re-stricted to A (defined in the following definition 3).
Definition 3. A map (a continuous-time system is de-fined similarly) has sensitive dependence on initial con-ditions when restricted to its invariant set A, if thereexists r, for any p0 ∈ A, and δ > 0, there exists p′0 ∈ A:|p′0 − p0| < δ, and an iterate k > 0 such that
|fk(p′0)− fk(p0)| > r. (22)
Attractor AL has already been taken as the smallestattracting set, so it is an indecomposable attractor bydefault. We just need to prove that the system has sensi-tive dependence on initial conditions when restricted toAL.We first prove that attractor AP of the Poincare map
is chaotic using the fact that the doubling map22 is sensi-tively dependent upon initial conditions when restrictedto its attractor. Then we prove in exactly the same waythat attractor AL of the model system is chaotic by thesensitivity of AP .
Proposition 1 (Sensitive Dependence of the PoincareMap). The Poincare map of the model system has sen-sitive dependence upon initial conditions when restrictedto its attractor AP .
Proof. For any p0 ∈ AP , with its neighboring initial pointp′0 ∈ AP , we take p′0 as (x′
0, y′
0) = (x′
0, y′
0 − δ · Sign(y0)).
Clearly, ‖pn − p′n‖ > |yn − y′n|. So, proving sensitivityto initial conditions of AP is equivalent to that of theDoubling Map:
yn+1 = ⌊2yn⌋, (yi ∈ (0, 1), ∀i). (23)
With the sensitive dependence upon initial conditions ofdoubling map when restricted to its attractor proved22,Poincare map of the model system is also sensitive whenrestricted to its attractor AP .
In exactly the same way, the model system can thusbe proved sensitively dependent upon initial conditionswhen restricted to its attractor AL. Hence, it is a chaoticattractor by definition.We also calculate the commonly used indicator of
chaos: Lyapunov exponents22 for the model system atfixed points. By solving the Lyapunov exponents in eachcoordinate direction, we find that in region RA: ℓx = 0,ℓy = 1, and ℓz = −1. It can be found that there is apositive Lyapunov exponent ℓy = 1 denoting exponen-tial expansion in the y direction. In region RB and RC ,ℓx = ℓy = ℓz = 0, which means that the expansion effectcausing the sensitivity of the system is mainly exerted inregion RA.From this and the former section, we find that the
attractor of the model system is a chaotic attractor withfractal dimension: a strange chaotic attractor36.
IV. CONSTRUCTION OF POTENTIAL FUNCTION IN
THE CHAOTIC SYSTEM
Based on the above observation, we want to startconstructing a potential function to describe the over-all structure of the chaotic system. First, we constructa “seed function”, F , to account for the “strangeness”of the system’s attractor. Then we prove its continuousdifferentiability so that it can be applied in the construc-tion of potential function for the model system. Later,we explicitly express the potential function in terms ofthe seed function F .
A. Definition of the “Seed Function” F
Definition 4 (Function F). Let
f1(x) =
{
0 , x ∈ [0 , 2/3]⋃
[4/3 , 2]1− cos(3πx) , x ∈ (2/3 , 4/3) ;
(24)
and
fn+1(x) =
1/9× fn(3x) , x ∈ [0 , 2/3]0 , x ∈ (2/3 , 4/3)1/9× fn(3x− 4) , x ∈ [4/3 , 2] .
(25)
Thus, we define the function F(x) as:
F(x) =
∞∑
n=1
fn(x). (26)
8
Figure 5 | Illustration of the Self-Similar Function:F. We hereby plot function F to intuitively visualize it.With F , construction of a potential function in the chaoticsystem would be natural.
Function F(x) defined on [0, 2] is shown in Figure (5).It has a fractal structure as the attractor AP of thePoincare map.
B. Proof of F(x) as Continuous Differentiable
Here, we propose that the function F(x) defined aboveis continuously differentiable.
Proposition 2 (Continuous Differentiability of FunctionF). Function F(x) =
∑
∞
n=1 fn(x) defined in Definition4 is continuously differentiable.
Proof. We first calculate the derivative of every fn(x) on[0, 2]. Then we bound fn(x) and |f ′
n(x)| by geometricseries to prove uniform convergence of their sums, andhence prove that F(x) =
∑
∞
n=1 fn(x) is continuously dif-ferentiable.From the definition (in equation (24) and (25)):
f1(x) =
{
0 , x ∈ [0 , 2/3]⋃
[4/3 , 2]1− cos(3πx) , x ∈ (2/3 , 4/3) ,
and
fn+1(x) =
1/9× fn(3x) , x ∈ [0 , 2/3]0 , x ∈ (2/3 , 4/3)1/9× fn(3x− 4) , x ∈ [4/3 , 2] ,
we have by taking derivative on both sides:
f ′
1(x) =
{
0 , x ∈ [0 , 2/3]⋃
[4/3 , 2]3π sin(3πx) , x ∈ (2/3 , 4/3) ,
and
f ′
n+1(x) =
1/3× f ′
n(3x) , x ∈ [0 , 2/3]0 , x ∈ (2/3 , 4/3)1/3× f ′
n(3x− 4) , x ∈ [4/3 , 2] .
Obviously, f1(x) 6 2 and |f ′
1(x)| 6 3π. And also,f ′
n(x) is continuous for any x ∈ [0, 2].If we further denote the set in which fn(x) is nonzero
as Cn, we have:
C1 =
(
2
3,4
3
)
,
and
Cn+1 =Cn
3
⋃ Cn + 4
3. (27)
Since Cn ⊂ [0, 2],
Cn
3
⋂ Cn + 4
3⊂
[
0,2
3
]
⋂
[
4
3, 2
]
= ∅.
We thus can conclude that:
fn+1(x) 61
9fn(x) 6 9−nf1(x) 6 2× 9−n,
and
|f ′
n+1(x)| 61
3|f ′
n(x)| 6 3−n|f ′
1(x)| 6 3π × 3−n,
for any x ∈ [0, 2].At this point, we give an upper bound for the series
∑mn=1 fn(x) and its derivative (Although the least up-
per bound is even smaller if we note that Cn
⋂
Cn+1 isactually empty, an upper bound is good enough):
m∑
n=1
fn(x) 69
4−
1
4× 9−m,
and that
m∑
n=1
|f ′
n(x)| 69π
2−
3π
2× 3−m.
So,∑
∞
n=1 f′
n(x) is uniformly absolutely-convergent.Hence,
∑
∞
n=1 f′
n(x) and∑
∞
n=1 fn(x) are all uniformlyconvergent.With every fn(x) continuously differentiable, F(x) =
∑
∞
n=1 fn(x) is continuously differentiable:
d
dxF(x) =
∞∑
n=1
f ′
n(x). (28)
Clearly, the points where F(x) = F ′(x) = 0 form aCantor set C corresponding to the attractor AL of themodel system (F(x) = 0 if and only if x ∈ C). At thispoint, we found that the potential function can be con-structed in the following section.
9
C. Constructing Potential Function in the Chaotic System
As in equation (11) we use θ to denote the angle that(y, z) form with respect to the center (2, 2):
θ = arccosy − 2
√
(z − 2)2 + (y − 2)2.
Then, we construct potential function in each region re-spectively:
1. In the right part of region RA, where x, y, z ∈ [0, 2],yz ∈ [−2, 2]:
ΨA =
(
4
9πθ +
5
9
)
F(x). (29)
2. In region RB, where x ∈ [0, 2], y ∈ (2, 4], z ∈ [0, 2],√
(z − 2)2 + (y − 2)2 ∈ [1, 2]:
ΨB =
(
4
9πθ +
5
9
)
F
(
(
4
3πθ +
1
3
)
−1
x
)
. (30)
3. In region RC , where x ∈ [0, 2/3], y ∈ [−2, 4], z > 2,√
(z − 2)2 + (y − 2)2 > 1,√
(z − 2)2 + (y − 3/2)2 6
5/2:
ΨC =
(
−4
9πθ +
5
9
)
F(3x). (31)
Here, we plot the potential function taken on thePoincare section in Figure (6).If we are also interested in the dynamics near saddle-
focus fixed points, potential function in region RDA ,RDB , and RDC can also be constructed as follows.
1. In region RDA , where x ∈ [0, 2], y ∈ [1, 2], z ∈ [1, 2],yz ∈ [2, 4]:
ΨDA =
(
4
9πθ +
5
9
)
F(x) + 1−
(
1
2yz − 2
)2
. (32)
2. In region RDB , where x ∈ [0 , 2/3 + 8/(3π)× θ], y ∈[2, 3], z ∈ [1, 2],
√
(z − 2)2 + (y − 2)2 ∈ [0, 1]:
ΨDB =
(
4
9πθ +
5
9
)
F
(
(
4
3πθ +
1
3
)
−1
x
)
(33)
+ 1− (z − 2)2 − (y − 2)2.
3. In region RDA , where x ∈ [0, 2/3], y ∈ [1, 3], z > 2,√
(z − 2)2 + (y − 2)2 6 1:
ΨDC =
(
−4
9πθ +
5
9
)
F(3x) (34)
+ 1− (z − 2)2 − (y − 2)2.
Figure 6 | Potential Function on Poincare Section.On the Poincare section, the potential function isdemonstrated to be a fractal object: it is zero when point(x, y, 2) belongs to the attractor AL, i.e., Ψ|z=2 = 0 whenx ∈ C. And when (x, y, 2) does not belong to the attractor,the potential function Ψ has self-similar structure.
Potential function in region RD is gradually higher in thecenter of the region than in its boundary with other re-gions. Hence, points in this region will naturally convergeto its boundary (at: x = 0,
√
(z − 2)2 + (y − 2)2 = 1,when y > 2 or z > 2; and at: x = 0, yz = 2, wheny, z ∈ [1, 2]).
The construction of the potential function is notunique. If we change the expression of f1(x) in the defini-tion of the “seed function” F(x), we can have a differentpotential function for the system.
V. VERIFICATION OF THE POTENTIAL FUNCTION
Here, in this section, we verify the integrity of the po-tential function from three angles: First, we show thatit is continuous in the domain. Second, we demonstratethat it decreases monotonically along the vector field.Third, we show that ∇Ψ(x) = 0 if and only if x belongsto the attractor of the system.
A. Continuity of the Potential Function
1. At the boundary of region RA and RB, y = 2, x ∈[0, 2], z ∈ [0, 1].
Hence, in equation (29), θ = arccos 0 = π/2;
(4/(3π)× θ + 1/3)−1
= 1; and (4/(9π)× θ + 5/9) =
10
7/9. So,
ΨA|y=2− =7
9F(x) = ΨB1
|y=2+ .
2. At the boundary of region RB and RC , z = 2, x ∈[0, 2/3], and y ∈ [3, 4].
Hence, in equation (30), θ = arccos1 = 0;
(4/(3π)× θ + 1/3)−1
= 3; and (±4/(9π)× θ + 5/9) =5/9. So,
ΨB|z=2− =5
9F(3x) = ΨC1
|z=2+ .
3. At the boundary of region RC and RA, z = 2, x ∈[0, 2/3], and y ∈ [−1, 1].
Hence, in equation (31), θ = arccos(−1) = π;(4/(9π)× θ + 5/9) = 1; and (−4/(9π)× θ + 5/9) =1/9. So,
ΨA|z=2− = F(x)
ΨC |z=2+ =1
9F(3x).
It can be observed that in the definition of F(x) =∑
∞
n=1 fn(x),
fn+1(x) =1
9fn(3x) , x ∈ [0 , 2/3] .
Clearly, because f1(x) = 0 in [0, 2/3],
1
9F(3x) =
∞∑
n=2
fn(x) = F(x), (35)
for x ∈ [0, 2/3].
Please note that this is a critical point in the construc-tion of potential function in cases of self-similar attrac-tors. If the potential function constructed is not self-similar accordingly, then the boundary would not totallyfit.
B. Monotonic Decreasing of the Potential Function
Then we take Lie derivative (derivative along the vec-tor field)37 of the potential function along vector fields ineach region remembering that F(x) > 0 for any x ∈ [0, 2].
1. In the right part of region RA, where x, y, z ∈ [0, 2],yz ∈ [−2, 2]:
x = 0
θ = Sgn(z − 2)(
−z − 2
(z − 2)2 + (y − 2)2y +
y − 2
(z − 2)2 + (y − 2)2z
)
= −(2− y)z + (2− z)y
(z − 2)2 + (y − 2)2.
Since x = 0 and y, z ∈ [0, 2],
ΨA =4
9πF(x)θ
= −4
9π
(2− y)z + (2− z)y
(z − 2)2 + (y − 2)2F(x) 6 0. (36)
2. In region RB, where x ∈ [0, 2], y ∈ (2, 4], z ∈ [0, 2],√
(z − 2)2 + (y − 2)2 ∈ [1, 2]:
x = −x
(
π
4+ arccos
y − 2√
(z − 2)2 + (y − 2)2
)
−1
= −x(π
4+ θ)
−1
θ =z − 2
(z − 2)2 + (y − 2)2y −
y − 2
(z − 2)2 + (y − 2)2z
= −1.
ΨB =4
9πF
(
(
4
3πθ +
1
3
)
−1
x
)
θ
+
(
4
9πθ +
5
9
)(
4
3πθ +
1
3
)
−2
F ′
(
(
4
3πθ +
1
3
)
−1
x
)
((
4
3πθ +
1
3
)
x−4
3πxθ
)
= −4
9πF
(
(
4
3πθ +
1
3
)
−1
x
)
+
(
4
9πθ +
5
9
)(
4
3πθ +
1
3
)
−2
F ′
(
(
4
3πθ +
1
3
)
−1
x
)
(
−
(
4
3πθ +
1
3
)
(π
4+ θ)
−1
x+4
3πx
)
.
Since
(
−
(
4
3πθ +
1
3
)
(π
4+ θ)
−1
x+4
3πx
)
= 0,
ΨB = −4
9πF
(
(
4
3πθ +
1
3
)
−1
x
)
6 0. (37)
3. In region RC , where x ∈ [0, 2/3], y ∈ [−1, 4], z > 2,√
(z − 2)2 + (y − 2)2 > 1,√
(z − 2)2 + (y − 3/2)2 6
5/2:
x = 0
θ = −z − 2
(z − 2)2 + (y − 2)2y +
y − 2
(z − 2)2 + (y − 2)2z
=(z − 2)2
(z − 2)2 + (y − 2)2+
y − 2
(z − 2)2 + (y − 2)2(
9y
8−
21
8+
√
(3y − 7)2 + 8(z − 2)2
8
)
> 0.
Hence,
ΨC = −4
9πF(3x)θ 6 0. (38)
11
C. Potential Function and the Attractor
Here, we verify that the potential function attains ex-tremum: ∇Ψ(x) = 0 if and only if x = (x, y, z) be-longs to the attractor AL of the system. Again, θ =arccos (y − 2)/
√
(z − 2)2 + (y − 2)2.
1. In the right part of region RA, where x, y, z ∈ [0, 2],
yz ∈ [−2, 2]:
∇ΨA =
(
4
9πθ +
5
9
)
F ′(x)
4
9π
z − 2
(z − 2)2 + (y − 2)2F(x)
−4
9π
y − 2
(z − 2)2 + (y − 2)2F(x)
.
Here, F(x) = 0 and F ′(x) = 0 if and only if x ∈ C; andpoint (x, y, z) belongs to the attractor AL if and onlyif x ∈ C. So, ∇ΨA = 0 if and only if (x, y, z) ∈ AL.
2. In region RB, where x ∈ [0, 2], y ∈ (2, 4], z ∈ [0, 2],√
(z − 2)2 + (y − 2)2 ∈ [1, 2]:
∇ΨB =
(
4
9πθ +
5
9
)(
4
3πθ +
1
3
)
−1
F ′
(
(
4
3πθ +
1
3
)
−1
x
)
4
9π
z − 2
(z − 2)2 + (y − 2)2
(
F
(
(
4
3πθ +
1
3
)
−1
x
)
−
(
4
3πθ +
5
3
)(
4
3πθ +
1
3
)
−2
F ′
(
(
4
3πθ +
1
3
)
−1
x
))
−4
9π
y − 2
(z − 2)2 + (y − 2)2
(
F
(
(
4
3πθ +
1
3
)
−1
x
)
−
(
4
3πθ +
5
3
)(
4
3πθ +
1
3
)
−2
F ′
(
(
4
3πθ +
1
3
)
−1
x
))
.
Here, F(
(4/(3π)× θ + 1/3)−1 x)
= 0 and F ′
(
(4/(3π)× θ + 1/3)−1 x)
= 0 if and only if (4/(3π)× θ + 1/3)−1 x ∈
C; and point (x, y, z) belongs to the attractor AL if and only if (4/(3π)× θ + 1/3)−1
x ∈ C. So, ∇ΨB = 0 if andonly if (x, y, z) ∈ AL.
3. In region C, where x ∈ [0, 2/3], y ∈ [−1, 4], z > 2,√
(z − 2)2 + (y − 2)2 > 1,√
(z − 2)2 + (y − 3/2)2 6
5/2:
∇ΨC =
(
−4
3πθ +
5
3
)
F ′(3x)
4
9π
z − 2
(z − 2)2 + (y − 2)2F(3x)
−4
9π
y − 2
(z − 2)2 + (y − 2)2F(3x)
.
Here, F(3x) = 0 and F ′(3x) = 0 if and only if x ∈ C;and point (x, y, z) belongs to the attractor AL if andonly if x ∈ C. So, ∇ΨC = 0 if and only if (x, y, z) ∈AL.
In exactly the same way, can we also show the integrityof the potential function in region RD.
VI. COMPARISON WITH RELATED WORKS
As discussed in the introduction, constructing apotential-like function in the chaotic system is actually aneffort that is by no means totally strange to researchers.Until recently, there are various efforts seeking to de-scribe chaotic dynamics using generalized Hamiltonian
approach14, energy-like function technique15, minimumaction method16, and etc. These previous methods allconstruct a potential-like scaler function to analyze cer-tain chaotic system. Unfortunately, the scalar functionsin these works all lack certain important properties.
For example, the generalized Hamiltonian systems ap-proach takes a quadratic form of the state variables asthe “generalized Hamiltonian”14, a Hamiltonian that in-cludes conserved dynamics, energy dissipation, and en-ergy input. The third part transforms an autonomousdifferential equation into a non-autonomous physicalmodel, attempting to explain for the “irregular”17 be-havior of chaotic systems. Contrary to this expectation,when the Hamiltonian is set possible to dissipate andincrease, the generalized Hamiltonian itself becomes achaotic oscillating signal with respect to time. There-fore, it remains an issue as to what additional insight thisgeneralized Hamiltonian can provide about the originalsystem, such as global stability or local performance.
The energy-like function technique is essentially simi-lar to the generalized Hamiltonian approach. Its energy-like function differs from the generalized Hamiltonian ina way that it may not be a quadratic form of the statevariables. Rather, the energy-like function is constructedbased on the “geometric appearance”15 of the attractorcorresponding to the specific chaotic system. Althoughthis technique would seem more sophisticated, its energy-
12
Figure 7 | Strange Chaotic Attractor.We find that a connected surface of the attractor is ofinfinite layers. We show how surface x = 2 is linked to theother equipotential layers. It can be seen that the surface ofthe attractor is orientable. We hereby demonstrate thestrange chaotic attractor viewed from (a), front; (b), side;and (c), top. The trajectory running from point (2, 1/4, 2)is also shown in the figure.
like function still oscillate chaotically along with time,describing chaotic dynamics in a chaotic fashion. Lossof monotonicity restricts the function from describingthe system’s essential properties like stability and per-formance.
The minimum action method, however, cast the prob-lem under the light of zero noise limit. By construct-ing an auxiliary Hamiltonian38 (commonly denoted as“Freidlin-Wentzell Hamiltonian”), Freidlin-Wentzell ac-tion functional can be minimized39. This method ana-lyzes chaotic system by possible transitions between limitsets16. But since the Freidlin-Wentzell Hamiltonian canbe not bounded even in globally stable systems, it is not aquantitative measure comparable between points in statespace, hence, not an ideal potential function.
In short, all the previous works each focuses on oneattribute of the potential function. However, as we cansee from our constructive result, only when all the re-quirements (in definition 1) are met, would the potentialfunction reflect evolution of the whole system and struc-ture of the chaotic attractor. In this sense, the currentwork is the first construction to satisfy such strong con-ditions, providing a both detailed and global descriptionfor a chaotic system.
VII. CHAOTIC ATTRACTOR AND STRANGE
ATTRACTOR
With the potential function constructed, we can easilysolve the system’s attractor without any need of numeri-cal simulation. We find that the attractor is composed ofconnected surfaces, each of infinite layers. Starting fromthe plane x = 2, we show the configuration of these layersin Figure (7). Since geometric configuration of chaotic at-tractor interests many researchers40, we demonstrate inthe figure that the chaotic attractor of the system studiedin this paper consists of orientable surfaces.The chaotic attractor here is a strange attractor of frac-
tal dimension36. And in the literature of dynamical sys-tems, there have long been discussions about the relation-ship between chaotic attractors and strange attractors41.Several examples of strange nonchaotic attractors andnonstrange chaotic attractors have been found36. Untilrecently, strange nonchaotic attractors are still studied42.The potential function approach provides a unified
framework to treat the topics of chaotic attractors andstrange attractors together. To clarify this insight, weneed to apply our decomposition method (equation (3))here:
x = f(x) = −D∇Ψ(x) +Q∇Ψ(x).
where
D = −f · ∇Ψ
∇Ψ · ∇ΨI,
and
Q =f ×∇Ψ
∇Ψ · ∇Ψ.
We first analyze our model system with this decompo-sition framework. Then we further modify our model sys-tem to two typical cases interesting to many researchers:a nonstrange chaotic attractor and a strange nonchaoticattractor. After analyzing these two cases, we explainthe different origins of chaotic attractors and strange at-tractors in general.
A. Decomposition of the Chaotic System
According to our decomposition scheme, we first de-compose the chaotic dynamical system in each regioninto two components: the gradient component and therotation component.In region RA, ∇ΨA is solved as:
∇ΨA =
(
4
9πθ +
5
9
)
F ′(x)
−4
9π
2− z
(z − 2)2 + (y − 2)2F(x)
4
9π
2− y
(z − 2)2 + (y − 2)2F(x)
.
13
Hence, we can find the expression of the matrix DA
accounting for the gradient component of the vector fieldin region RA: DA = −
fA · ∇ΨA
∇ΨA · ∇ΨA
I (39)
=
4
9π
(2− y)z + (2− z)y
(z − 2)2 + (y − 2)2F(x)
(
4
9πθ +
5
9
)2
(F ′(x))2 +(4/9π)2
(z − 2)2 + (y − 2)2(F(x))2
I.
The decomposed gradient part would then be:
DA∇ΨA =
4
9π
(2− y)z + (2− z)y
(z − 2)2 + (y − 2)2F(x)
(
4
9πθ +
5
9
)2
(F ′(x))2 +(4/9π)
2
(z − 2)2 + (y − 2)2(F(x))2
(
4
9πθ +
5
9
)
F ′(x)
−4
9π
2− z
(z − 2)2 + (y − 2)2F(x)
4
9π
2− y
(z − 2)2 + (y − 2)2F(x)
.
Also, we can find the decomposed rotation part by finding QA as:
QA =fA ×∇ΨA
∇ΨA · ∇ΨA
(40)
=1
(
4
9πθ +
5
9
)2
(F ′(x))2+
(4/9π)2
(z − 2)2 + (y − 2)2(F(x))
2
0 −y
(
4
9πθ +
5
9
)
F ′(x) z
(
4
9πθ +
5
9
)
F ′(x)
y
(
4
9πθ +
5
9
)
F ′(x) 0 −4
9π
(2− z)z − (2− y)y
(z − 2)2 + (y − 2)2F(x)
−z
(
4
9πθ +
5
9
)
F ′(x)4
9π
(2− z)z − (2− y)y
(z − 2)2 + (y − 2)2F(x) 0
.
The decomposed rotation part would then be:
QA∇ΨA =1
(
4
9πθ +
5
9
)2
(F ′(x))2+
(4/9π)2
(z − 2)2 + (y − 2)2(F(x))
2
(41)
−4
9π
(
4
9πθ +
5
9
)
(y − 2)z + (z − 2)y
(z − 2)2 + (y − 2)2F ′(x)F(x)
y
(
4
9πθ +
5
9
)2
(F ′(x))2+
(
4
9π
)2(y − 2)2y − (y − 2)(z − 2)z
((z − 2)2 + (y − 2)2)2 (F(x))
2
−z
(
4
9πθ +
5
9
)2
(F ′(x))2+
(
4
9π
)2(y − 2)y(z − 2)− (z − 2)2z
((z − 2)2 + (y − 2)2)2 (F(x))
2
.
When the system approaches its attractor, i.e., F(x) → 0,
F(x)
(F ′(x))2 = lim
x→0
(
1
9
)n
(1− cos(3πx))
((
1
3
)n
3π sin(3πx)
)2 (42)
= limx→0
(
1
9
)n1
2(3πx)2
((
1
3
)n
9π2x
)2 =1
18π2.
14
Thus, when the system converges to its attractor, thegradient matrix DA would be:
DA = −
4
9π
(y − 2)z + (z − 2)y
(z − 2)2 + (y − 2)2(
4
9πθ +
5
9
)2
F(x)
(F ′(x))2I (43)
= −2
81π3
(y − 2)z + (z − 2)y
(z − 2)2 + (y − 2)2(
4
9πθ +
5
9
)2 I,
which is finite.Hence, the gradient component DA∇ΨA of the system
would converges to zero when approaching the attractor.So the motion on the attractor is caused totally by therotation part: QA∇ΨA.In exactly the same way, the decomposition procedure
can be carried out in region B and region C, and the sameconclusion holds.
B. Nonstrange Chaotic Attractor
Let’s first examine an example of nonstrange chaoticattractor by modifying our original system a little (inregion RB, equation (11)):In region RB, we set
θ = arccosy − 2
√
(z − 2)2 + (y − 2)2,
as in equation (11). Then we change the dynamical sys-tem in region RB (defined as x ∈ [0 , 4/π× θ], y ∈ [2, 4],
z ∈ [0, 2],√
(z − 2)2 + (y − 2)2 ∈ [1, 2]) to:
x = −x
θy = 2− zz = y − 2.
The same as in the original system, domain of definitioncan be expanded to the whole R3 space.Consequently, the Poincare map would be as follows:When (x, y) ∈ [0, 2]× [0, 1],
{
xn+1 = 0yn+1 = 2yn − 1.
When (x, y) ∈ [0, 2]× [−1, 0),
{
xn+1 = 1yn+1 = 2yn + 1.
The attractor A′
L of the modified system would be:
(assuming θ = arccos (y − 2)/√
(z − 2)2 + (y − 2)2):In region RA, x = 0 or 2;in region RB, (π/2)× (x/θ) = 0 or 2;
in region RC , x = 0 or 2.The attractor is shown in Figure (8). We can calculate
its box-counting dimension to be:
db(A′
L) = limǫ→0
logN(ǫ,A′
L)
log(1/ǫ)= 2, (44)
which is an integer dimension. Actually, the attractor A′
L
is just two orientable surfaces folded together. Hence, itis no longer a strange attractor anymore.Exactly as in the original system, the modified attrac-
tor can be proved to be chaotic. And we can also cal-culate the commonly used indicator of chaos: Lyapunovexponents22 for the model system at fixed points. Lya-punov exponents are solved in each direction as: ℓx = 0,ℓy = 1, and ℓz = −1 in region RA (In other regions,ℓx = ℓy = ℓz = 0). It is found that there is a positiveLyapunov exponent ℓy = 1 denoting exponential expan-sion in the y direction, exactly as in the original modelsystem.So, it is clear that the modified attractor is a non-
strange chaotic attractor.Now, we construct a potential function Φ for the new
dynamical system by first appointing a new seed functionF (x) defined in [0, 2]:
F (x) = 1− cos(πx) , x ∈ [0 , 2] .
Hence, the potential function can be represented as:
1. In the right part of region RA, where x, y, z ∈ [0, 2]:
ΦA =
(
θ
π
)
F (x).
2. In region RB , where y ∈ [2, 4], z ∈ [0, 2],√
(z − 2)2 + (y − 2)2 ∈ [1, 2], x ∈ [0 , 4/π × θ]:
ΦB =
(
θ
π
)
F(πx
2θ
)
.
3. In region RC , where x = 2, y ∈ [−2, 4], z > 2,√
(z − 2)2 + (y − 2)2 > 1,√
(z − 2)2 + (y − 3/2)2 6
5/2:
ΦC = 0.
Here, Φ = 0 corresponds to the attractor.We can further decompose the system as with the orig-
inal model system:
x = f(x) = −D∇Φ(x) +Q∇Φ(x).
Then, in region RA:
∇ΦA =1
π
θF ′(x)
−2− z
(z − 2)2 + (y − 2)2F (x)
2− y
(z − 2)2 + (y − 2)2F (x)
.
15
DA = −fA · ∇ΦA
∇ΦA · ∇ΦA
I (45)
=
π(2 − y)z + (2 − z)y
(z − 2)2 + (y − 2)2F (x)
θ2 (F ′(x))2+
1
(z − 2)2 + (y − 2)2(F (x))
2I.
The decomposed gradient part would then be:
DA∇ΦA =
(2− y)z + (2 − z)y
(z − 2)2 + (y − 2)2F (x)
θ2 (F ′(x))2+
(F (x))2
(z − 2)2 + (y − 2)2
(46)
θF ′(x)
−2− z
(z − 2)2 + (y − 2)2F (x)
2− y
(z − 2)2 + (y − 2)2F (x)
.
Also, we can find the decomposed rotation part by finding QA as:
QA =fA ×∇ΦA
∇ΦA · ∇ΦA
(47)
=π
θ2 (F ′(x))2+ (F (x))
2
0 −yθF ′(x) zθF ′(x)
yθF ′(x) 0 −(2− z)z − (2− y)y
(z − 2)2 + (y − 2)2F (x)
−zθF ′(x)(2 − z)z − (2− y)y
(z − 2)2 + (y − 2)2F (x) 0
.
The decomposed rotation part would then be:
QA∇ΦA =π
θ2 (F ′(x))2+ (F (x))
2 (48)
−(y − 2)z + (z − 2)y
(z − 2)2 + (y − 2)2θF ′(x)F (x)
yθ2 (F ′(x))2+
(y − 2)2y − (y − 2)(z − 2)z
((z − 2)2 + (y − 2)2)2 (F (x))
2
−zθ2 (F ′(x))2+
(y − 2)y(z − 2)− (z − 2)2z
((z − 2)2 + (y − 2)2)2 (F (x))
2
.
The properties of the gradient part and the rotationpart corresponds exactly to the original model system.That is: the gradient part converges to zero when ap-proaching the attractor; motion on the attractor is de-termined by the rotation part.
C. Strange Nonchaotic Attractor
A strange nonchaotic attractor can also be con-structed.
We simply take the gradient of potential function Ψ ofthe original system in each region of definition:
x = −∂xΨy = −∂yΨz = −∂zΨ.
If we take left part of regionRA for example, the vector
16
Figure 8 | Nonstrange Chaotic Attractor.We change the expression of the system a little, so that theattractor is just two orientable surfaces folded together,rather than a fractal structure. However, it remains to be achaotic attractor. We hereby demonstrate the nonstrangechaotic attractor viewed from (a), front; (b), side; and (c),top. The trajectory running from point (2, 1/4, 2) is alsoshown in the figure.
field would be:
fA =
−
(
4
9πθ +
5
9
)
F ′(x)
−4
9π
z − 2
(z − 2)2 + (y − 2)2F(x)
4
9π
y − 2
(z − 2)2 + (y − 2)2F(x)
.
The resultant ODE system defined by the gradient is adynamical since it is Lipschitz continuous in each region.And with existence and uniqueness of the flow guaran-teed by Lipschitz continuity, conditions for the systembeing a dynamical system can be satisfied and extendedto include boundaries.The system would converge downward the potential
function Ψ until reaching the states where Ψ = 0. Con-sequently, attractor of this system is characterized byΨ = 0, as in the original model system. Hence, the gra-dient system’s attractor is the same attractor AL of theoriginal model system, whose box-counting dimension:
db(AL) = limǫ→0
logN(ǫ,AL)
log(1/ǫ)= 2 + ln(2)/ln(3). (49)
Hence, the system has a strange attractor.Since ∇Ψ = 0 when Ψ = 0, the dynamical system is
not sensitively dependent upon initial conditions whenrestricted to the attractor. So, the attractor is notchaotic. Also, its Lyapunov exponents at the fixed points(where x ∈ C, y = z = 0) would be: ℓx = −(1/3)n× 9π2,ℓy = ℓz = 0. Hence, it’s a strange nonchaotic attractor.
Decomposition of this system would give: D(x) = Iand Q(x) = 0. Thus, D∇Ψ(x) = −f(x) is just the re-versed gradient system.
D. Chaotic Attractor versus Strange Attractor
The previous two examples show that the concepts ofchaotic attractor and strange attractor do not imply eachother. Under our framework of decomposition (equation(3)) here:
x = f(x) = −D∇Ψ(x) +Q∇Ψ(x).
Since Ψ = ∇Ψ · x = ∇Ψ · (D∇Ψ), Ψ decreases mono-tonically according to the gradient component D∇Ψ ofthe vector field f . Then the attractor is naturally char-acterized by D∇Ψ = 0. So, whether the attractor is astrange attractor is determined by the gradient part ofthe vector field.Sensitive dependence upon initial conditions when re-
stricted to the attractor, however, is determined by therotation part of the vector field: Q∇Ψ. Once the sys-tem has evolved to the limit set, D∇Ψ would equal tozero, and Q∇Ψ would be prevalent. Hence, the rota-tional vector field on the attractor causes the expansionof the state space, leading to dynamical sensitivity. Con-versely, when Q∇Ψ = 0, chaotic motion on the attractorwould not exist. In this sense, nonzero rotation part ofthe dynamical system is a necessary condition for causinghyperbolic chaos34.To sum up, gradient part and rotation part of the vec-
tor field are responsible for the creation of strange at-tractor and chaotic attractor correspondingly. Althoughthey are both affected by the geometrical configurationof the potential function Ψ, they denote dissipation andcirculation respectively.
VIII. CONCLUSION
In the present paper, it is shown that potential func-tions with monotonic properties can be constructed incontinuous dissipative chaotic systems with strange at-tractors. The potential function here is a continuousfunction in phase space, monotonically decreasing withtime and remains constant if and only if limit set isreached. This definition is a natural restriction of genericdynamics since it is a direct generalization of Lyapunovfunction and corresponds to the concept of energy.Potential function defined this way also implies that
the dynamics can be decomposed into two parts: a gra-dient part, dissipating energy potential; and a rotationpart, conserving energy potential. The gradient partdrives the system towards the attractor while the rota-tion part perpetuates the system’s circular motion on theattractor.To demonstrate the power of this framework in chaotic
systems, we simplify the geometric Lorenz attractor, and
17
prove by definition that it is a chaotic attractor. Then weanalytically and explicitly construct a suitable potentialfunction for the attractor, which, to our best knowledge,is the first example in chaotic dynamics. The potentialfunction reveals the fractal nature of the chaotic strangeattractor.We further analyze the concept of chaotic attractor and
strange attractor with our decomposition. It is foundthat chaotic attractor originates in the rotation part,prompting the state space of the attractor to expand;while strange attractor originates in the gradient part,causing initial states attracted to complex limit set.
ACKNOWLEDGEMENTS
The authors would like to express their sincere grati-tude to Xinan Wang, Ying Tang, Song Xu and JianghongShi for their constructive advice throughout this work.The authors also appreciate valuable discussion withJames A. Yorke, David Cai, and Shijun Liao.This work is supported in part by the Natural Science
Foundation of China No. NFSC61073087, the National973 Projects No. 2010CB529200, and the Natural ScienceFoundation of China No. NFSC91029738.
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